Two approximation algorithms for bipartite matching on multicore architectures

Abstract

To cite this version:Fanny Dufossé, Kamer Kaya, Bora Uçar. Two approximation algorithms for bipartite matching on multicore architectures. Journal of Parallel and Distributed Computing, Elsevier, 2015, 85, pp.62-78. 10.1016/j.jpdc.2015 Two approximation algorithms for bipartite matching on multicore architectures AbstractWe propose two heuristics for the bipartite matching problem that are amenable to shared-memory parallelization. The first heuristic is very intriguing from a parallelization perspective. It has no significant algorithmic synchronization overhead and no conflict resolution is needed across threads. We show that this heuristic has an approximation ratio of around 0.632 under some common conditions. The second heuristic is designed to obtain a larger matching by employing the well-known Karp-Sipser heuristic on a judiciously chosen subgraph of the original graph. We show that the Karp-Sipser heuristic always finds a maximum cardinality matching in the chosen subgraph. Although the Karp-Sipser heuristic is hard to parallelize for general graphs, we exploit the structure of the selected subgraphs to propose a specialized implementation which demonstrates very good scalability. We prove that this second heuristic has an approximation guarantee of around 0.866 under the same conditions as in the first algorithm. We discuss parallel implementations of the proposed heuristics on a multicore architecture. Experimental results, for demonstrating speed-ups and verifying the theoretical results in practice, are provided.

Approximation algorithms in combinatorial scientific computing

“…Other scaling algorithms could also be used for this purpose, but the Sinkhorn-Knopp algorithm is more concurrent. Also Dufossé, Kaya and Uçar (2015) report that five to ten iterations of this scaling algorithm suffice to compute approximate matchings.…”

“…In this algorithm a row could attempt to match to a column that is already matched to another row; in this case, one of the rows, say the last, succeeds, and the other row gets unmatched. Dufossé et al (2015) proved that this random matching will match n(1 − 1/e) vertices with high probability, where e is the base of natural logarithm, the Euler number.…”

Effective Heuristics for Matchings in Hypergraphs

“…In this case, the entries not participating in any perfect matching tend to zero in the scaled matrix. This fact is exploited to design randomized approximation algorithms for the maximum cardinality matching problem in graphs [12,13]. By scaling the adjacency matrix in a preprocess and choosing edges with a probability corresponding to the scaled value of the associated matrix entry, the edges which are not included in a perfect matching become less likely to be chosen.…”

“…To evaluate and emphasize the contribution of scaling better, we compare the [12]. Let A KS be an n × n matrix.…”

“…Karp-Sipser introduces certain rules to Greedy to improve the cardinality. The third heuristic is inspired by a recent scaling-based approach proposed for the maximum cardinality matching problem on graphs [11][12][13]. The fourth heuristic is a modification of the third one that allows for faster execution time.…”